1. **Problem Statement:** We are given 15 samples, each with 10 measurements of soft-drink fill volumes. We need to set up the $\bar{x}$ and $R$ charts, check for statistical control, and compare the $R$ chart with the $s$ chart. 2. **Formulas and Important Rules:** - Sample mean: $\bar{x}_i = \frac{1}{n} \sum_{j=1}^n x_{ij}$ - Sample range: $R_i = \max(x_{ij}) - \min(x_{ij})$ - Sample standard deviation: $s_i = \sqrt{\frac{1}{n-1} \sum_{j=1}^n (x_{ij} - \bar{x}_i)^2}$ - Control limits for $\bar{x}$ chart: $$UCL_{\bar{x}} = \bar{\bar{x}} + A_2 \bar{R}, \quad LCL_{\bar{x}} = \bar{\bar{x}} - A_2 \bar{R}$$ - Control limits for $R$ chart: $$UCL_R = D_4 \bar{R}, \quad LCL_R = D_3 \bar{R}$$ - Constants for $n=10$: $A_2=0.308$, $D_3=0.223$, $D_4=1.777$ 3. **Calculate Sample Means ($\bar{x}_i$), Ranges ($R_i$), and Standard Deviations ($s_i$):** | Sample | $\bar{x}_i$ | $R_i$ | $s_i$ | |--------|-------------|-------|-------| | 1 | 0.6 | 4.0 | 1.58 | | 2 | 0.65 | 2.5 | 0.93 | | 3 | 0.0 | 2.5 | 1.11 | | 4 | -0.3 | 3.5 | 1.22 | | 5 | 0.0 | 1.5 | 0.71 | | 6 | 0.1 | 3.0 | 1.12 | | 7 | 0.1 | 2.5 | 0.99 | | 8 | -0.1 | 3.0 | 0.87 | | 9 | 0.25 | 3.0 | 1.11 | | 10 | -0.15 | 5.0 | 1.72 | | 11 | 0.7 | 2.5 | 1.05 | | 12 | 0.05 | 4.0 | 1.18 | | 13 | -0.7 | 2.5 | 0.75 | | 14 | 0.0 | 3.5 | 1.22 | | 15 | 0.3 | 3.0 | 1.22 | 4. **Calculate Overall Averages:** - $\bar{\bar{x}} = \frac{1}{15} \sum_{i=1}^{15} \bar{x}_i = \frac{2.75}{15} = 0.1833$ - $\bar{R} = \frac{1}{15} \sum_{i=1}^{15} R_i = \frac{46.5}{15} = 3.1$ 5. **Calculate Control Limits:** - $UCL_{\bar{x}} = 0.1833 + 0.308 \times 3.1 = 0.1833 + 0.9548 = 1.1381$ - $LCL_{\bar{x}} = 0.1833 - 0.9548 = -0.7715$ - $UCL_R = 1.777 \times 3.1 = 5.509$ - $LCL_R = 0.223 \times 3.1 = 0.691$ 6. **Check for Statistical Control:** - All $\bar{x}_i$ values lie between $-0.7715$ and $1.1381$. - All $R_i$ values lie between $0.691$ and $5.509$. - No points outside control limits, so the process appears in statistical control. 7. **Revised Control Limits (if needed):** - Since no points are outside limits, no revision needed. 8. **Comparison of $R$ and $s$ Charts:** - $R$ chart uses range, simpler but less sensitive. - $s$ chart uses standard deviation, more sensitive to variability. - Both charts show consistent variability patterns here. **Final conclusion:** The process is in statistical control based on $\bar{x}$ and $R$ charts. The $s$ chart confirms variability patterns similar to the $R$ chart.